How can I twist part of my mesh into a Möbius strip shape?
Rather easy to make.


Select an edge loop that is running across the width of the circle ridge, with AltRMB , then press V to rip it, RMB  or Esc to cancel the transform. (You now have eight vertices in the same spot where the original four use to be.) Don't change your selection
Turn on Connected Proportional Editing AltO, or in the 3D view header use the Proportional Editing button.
Press RY 180 now before confirming the rotation use the mouse wheel to change the falloff of the proportional editing. You do not what too many other faces to turn, but you need a few.
I found it looked better when I moved the edges in the turn up a little bit. It Keeps the twist more balanced.

AA (to select everything) then W > Remove Doubles because you still have two sets of edges in that original spot.


If you've gotten this far, the modeling should be done, but it will look really bad.


A Subsurf modifier is all that is needed to fix it, but first set the Mean Crease to one on all the edge loops along the curve. (The object in the image in your question already has this, so you can skip this step.) You can quickly select them all in edge select mode with AltRMB  and holding shift for each one after the first. Then in the properties panel (N) under the Transform section you will find the Mean Crease value.

The whole process is in the gif below.